1 whole chicken (about 3 1/2 pound), backbone discarded and chicken cut French style into 8 pieces (see cooks' note, below)
1 tablespoon vegetable oil
3 tablespoons unsalted butter, divided
4 medium leeks (white and pale green parts only), finely chopped (2 cups)
2 tablespoons finely chopped shallot
4 medium carrots, halved diagonally
1 cup dry white wine (preferably Alsatian Riesling)
1 1/2 pound small (2-inch) red potatoes
2 tablespoons finely chopped flat-leaf parsley
1/2 cup crème fraîche or heavy cream
Fresh lemon juice to taste